In this paper, we consider a class of convection-diffusion equations with memory effects. These equations arise as a result of homogenization or upscaling of linear transport equations in heterogeneous media and play an important role in many applications. First, we present a dememorization technique for these equations. We show that the convection-diffusion equations with memory effects can be written as a system of standard convection-diffusion-reaction equations. This allows removing the memory term and simplifying the computations. We consider a relation between dememorized equations and micro-scale equations, which do not contain memory terms. We note that dememorized equations differ from micro-scale equations and constitute a macroscopic model. Next, we consider both implicit and partially explicit methods. The latter is introduced for problems in multiscale media with high-contrast properties. Because of high contrast, explicit methods are restrictive and require time steps that are very small (scales as the inverse of the contrast). We show that, by appropriately decomposing the space, we can treat only a few degrees of freedom implicitly and the remaining degrees of freedom explicitly. We present a stability analysis. Numerical results are presented that confirm our theoretical findings of partially explicit schemes applied to dememorized systems of equations.

Blazars, particularly Flat Spectrum Radio Quasars (FSRQs), are well-known for their ability to accelerate a substantial population of electrons and positrons, as inferred from multiwavelength radiation observations. Therefore, these astrophysical objects are promising candidates for studying high-energy electron--positron interactions, such as the production of $W^{\pm}$ and $Z$ bosons. In this work, we explore the implications of electron--positron annihilation processes in the jet environments of FSRQs, focusing on the resonant production of electroweak bosons and their potential contribution to the diffuse neutrino flux. By modeling the electron distribution in the jet of the FSRQ 3C~279 during a flaring state, we calculate the reaction rates for $W^{\pm}$ and $Z$ bosons and estimate the resulting diffuse fluxes from the cosmological population of this http URL incorporate the FSRQ luminosity function and its redshift evolution to account for the population distribution across cosmic time, finding that the differential flux contribution exhibits a pronounced peak at redshift $z \sim 1$. While the expected fluxes remain well below the detection thresholds of current neutrino observatories such as IceCube, KM3NeT, or Baikal-GVD, the expected flux from the $Z$ boson production could account for approximately $10^{-3}$ of the total diffuse astrophysical neutrino flux. These results provide a theoretical benchmark for the role of Standard Model electroweak processes in extreme astrophysical environments and emphasize the interplay between particle physics and astrophysics, illustrating that even rare high-energy interactions can leave a subtle but quantifiable imprint on the diffuse astrophysical neutrinos.

The least squares method provides the best-fit curve by minimizing the total squares error. In this work, we provide the modified least squares method based on the fractional orthogonal polynomials that belong to the space $M_{n}^{\lambda} := \text{span}\{1,x^{\lambda},x^{2\lambda},\ldots,x^{n\lambda}\},~\lambda \in (0,2]$. Numerical experiments demonstrate how to solve different problems using the modified least squares method. Moreover, the results show the advantage of the modified least squares method compared to the classical least squares method. Furthermore, we discuss the various applications of the modified least squares method in the fields like fractional differential/integral equations and machine learning.

The $s$-channel process $\bar\nu_ee^-\rightarrow W^-$(on-shell) is now referred to as the Glashow resonance and being searched for at kilometer-scale neutrino ice/water detectors like IceCube, Baikal-GVD or KM3NeT. After over a decade of observations, IceCube has recorded only a few relevant neutrino events such that further exploration yet remains necessary for unambiguous confirmation of the existence of this resonant interaction. Meanwhile, its experimental discovery would provide an additional important test of the Standard Model. One might therefore ask: are there reactions with the Glashow resonance that would not necessitate having initial (anti)neutrino beams? This article suggests a surprisingly positive answer to the question $-$ namely, that the process may proceed in electron-positron collisions at accelerator energies, occurring as $e^+e^-\rightarrow W^-\rho(770)^+$. Although the resonance appears somewhat disguised, the underlying physics is transparent, quite resembling the well known radiative return: emission of $\rho^+$ from the initial state converts the incident $e^+$ into $\bar\nu_e$. Likewise, the CP conjugate channel, $\nu_e e^+\rightarrow W^+$, takes the form $e^+e^-\rightarrow W^+\rho(770)^-$. Similar reactions with muons are also possible. Within this viewpoint, future high-luminosity lepton colliders seem to be promising for excitation of the Glashow resonance in laboratory conditions.

In homogenization theory, mathematical models at the macro level are constructed based on the solution of auxiliary cell problems at the micro level within a single periodicity cell. These problems are formulated using asymptotic expansions of the solution with respect to a small parameter, which represents the characteristic size of spatial heterogeneity. When studying diffusion equations with contrasting coefficients, special attention is given to nonlocal models with weakly conducting inclusions. In this case, macro-level processes are described by integro-differential equations, where the difference kernel is determined by the solution of a nonstationary cell problem. The main contribution of this work is the development of a computational framework for the homogenization of nonstationary processes, accounting for memory effects. The effective diffusion tensor is computed using a standard numerical procedure based on finite element discretization in space. The memory kernel is approximated by a sum of exponentials obtained from solving a partial spectral problem on the periodicity cell. The nonlocal macro-level problem is transformed into a local one, where memory effects are incorporated through the solution of auxiliary nonstationary problems. Standard two-level time discretization schemes are employed, and unconditional stability of the discrete solutions is proved in appropriate norms. Key aspects of the proposed computational homogenization technique are illustrated by solving a two-dimensional model problem.